CN106099921B

CN106099921B - A kind of Power System Delay stability margin fast solution method

Info

Publication number: CN106099921B
Application number: CN201610594776.XA
Authority: CN
Inventors: 贾宏杰; 董朝宇; 姜懿郎; 姜涛; 王蕾
Original assignee: Tianjin University
Current assignee: Tianjin University
Priority date: 2016-07-21
Filing date: 2016-07-21
Publication date: 2018-09-28
Anticipated expiration: 2036-07-21
Also published as: US20210286917A1; US11301598B2; WO2018014417A1; CN106099921A

Abstract

The invention discloses a kind of Power System Delay stability margin fast solution methods to reconstruct new time lag system, reduce system dimension using Jordan canonical transformations, the separation of Taylor series and three steps of Schur model reductions as core.This method is first handled time lag system model with Jordan canonical transformations；Further the thought of Taylor series expansions is applied in the separation process containing hangover state variable and without hangover state variable；Schur model order reducing methods are recycled to realize that equilibrium model blocks；Finally, proposed inventive method is verified under several typical criterions respectively using 3 machines of the WSCC of multiple time delay, 9 node power system.The present invention is greatly reduced the solution time of delay margin, completes the rapid solving of Power System Delay stability margin, it is difficult to solve the problems, such as that multidimensional, Systems with Multiple Time-Delays solve, expands the application range of stability criterion by simplifying system model.

Description

A kind of Power System Delay stability margin fast solution method

Technical field

The present invention relates to linear matrix inequality (LMI) technology and time lag system stability analysis fields more particularly to one Kind time lag system stability margin fast solution method.

Background technology

Wide Area Measurement System (Wide Area Measurement System, WAMS) is to be with synchronized phase measurement technology Basis with dynamic process of electrical power system detection, analysis and controls the real-time monitoring system for target.In wide area measurement system, The operation stability that wide-area control is conducive to improve entire electric system is carried out using wide area measurement information, realizes electric system Safe and stable operation.But in the electrical power system wide-area control based on WAMS, due to needing to be fed back using distant signal Control, there is certain time lag in wide area measurement signal, and the presence of time lag is possible to deteriorate controller performance, and system is caused to go out Existing unstability, therefore, influence of the research wide area measurement time lag to stability of power system have a very important significance.

Influence for research time lag to electric system, while considering that time lag random fluctuation, parameter are discontinuous and presence is cut Situations such as ring change section, the direct method based on Lyapunov Theory of Stability and LMI technologies have been widely used, it is by constructing energy Flow function can carry out time lag system controller design etc. with direct solution time lag system stability margin.

But when direct method to be applied to the analysis and processing of wide area system Time Delay, this method variable number to be solved It is approximate with the quantity of state variable in system in square, when state variable is more in system, solves time lag system and stablize Nargin will face that unknown variable is excessive, calculate the difficulty of overlong time, therefore how to reduce the number of unknown variable, reduce meter Calculating the solution time just becomes particularly important.There is the method for some model simplifications at present, such as：Equilibrium, which is blocked, utilizes root mean square pair Complex model is simplified；To non-minimum system, simplified system is approached with Hankel minimum degrees, and real with Coprime Factor Existing model blocks, these methods succeed application in complex dynamic systems analysis.However, above-mentioned simplified method is big Majority is directed to the system without time lag, surmounts item since the characteristic equation of time lag system contains, by existing simplified method The solution procedure for being applied directly to time lag stabilization still has bigger difficulty.

Invention content

To solve the above problems, the present invention proposes a kind of Power System Delay stability margin fast solution method.The party Method is by handling the system model containing time lag with Jordan canonical transformations；Further Taylor series is used for containing Hangover state variable and association separation process without hangover state variable；Schur model order reducing methods are recycled to realize balanced Model blocks, and the solution time of delay margin is greatly reduced, meanwhile, it ensure that the accuracy of last judging result, Under Layapunov stability criterias, it is steady that time lag is carried out to the 9 node power system of WSCC-3 machines of multiple time delay using the method for the present invention Determine nargin rapid solving.

A kind of Power System Delay stability margin fast solution method proposed by the present invention, for containing m Time Delay Time-lag power system, build time-lag power system mathematical model, using Jordan canonical transformations, Taylor series separation, Schur model reductions simplify the time-lag power system mathematical model, the time-lag power system mathematics after being simplified Model, and then the delay margin of time-lag power system is quickly found out, to be guaranteed, power system stability operation is permitted Perhaps maximum time lag, is as follows：

Step 1: structure time-lag power system mathematical model：

In formula：T indicates time variable；X (t) is state variable；It is state variable to the derivative of time；A₀When being non- Stagnant coefficient matrix；A_i, i=1,2 ..., m are time delay matrix, and m indicates Time Delay number；τ_i, i=1,2 ..., m, to be The time lag constant of system；τ_i>0 expression time lag is all higher than 0；x(t-τ_i), i=1,2 ..., m are hangover state variable；H (t, ξ) is The historical track of state variable x (t)；ξ∈[-max(τ_i), 0) indicate variable ξ in τ_iChange between the opposite number of maximum value and 0； Above-mentioned algebraic variable belongs to real number field R, and above-mentioned vector variable belongs to n dimension real vectors Rⁿ；

Step 2: utilizing time delay matrix A_iSparsity, to step 1 structure time-lag power system mathematical model In time delay matrix A_i, i=1, the sum of 2 ..., m progress Jordan canonical transformations, and according to sparsity to time lag power train Mathematical model procession of uniting converts；

Step 3: utilizing the hangover state variable after row-column transform in Taylor series expansion step 2Separation The non-time lag item in time-lag power system mathematical model after Jordan canonical transformations and sparsity row-column transformAnd when Stagnant itemBetween it is interrelated；

Step 4: the time-lag power system mathematical modulo using Schur models to step 3 after the separation of Taylor series Type is simplified, and the number of state variable is reduced to r by n, finally obtains the time-lag power system mathematical model after simplifying；

Step 5: under Power System Delay stability criteria, the time lag power train after the simplification obtained using step 4 System mathematical model finds out the delay margin of the time-lag power system containing m Time Delay, and it is steady to complete Power System Delay The rapid solving for determining nargin, to the permitted maximum time lag of the power system stability operation that is guaranteed.

Compared with prior art, the beneficial effects of the invention are as follows：

The inventive method proposes a kind of Power System Delay stability margin for the wide area power system containing multiple time delay Rapid solving (referred to as JTS rapid solvings) method, is dropped using Jordan canonical transformations, the separation of Taylor series, Schur models Rank realizes the model simplification to time-lag power system, and the model after simplifying can be suitable for various time lag stability criterias, can Greatly improve computational efficiency, change calculated performance.

Description of the drawings

Fig. 1 is the 9 node power system time lags inhibited stably figure of WSCC-3 machines of three kinds of criterion direct solutions；

Fig. 2 is the 9 node power system time lags of WSCC-3 machines that three kinds of criterions are solved using JTS fast solution methods of the present invention Inhibited stably figure.

Specific implementation mode

Technical solution of the present invention is described in further detail in the following with reference to the drawings and specific embodiments, it is described specific Embodiment is only explained the present invention, is not intended to limit the invention.

A kind of Power System Delay stability margin fast solution method proposed by the present invention, for containing m Time Delay Time-lag power system, build time-lag power system mathematical model, using Jordan canonical transformations, Taylor series separation, Schur model reductions simplify the time-lag power system mathematical model, the time-lag power system mathematics after being simplified Model, and then the delay margin of time-lag power system is quickly found out, judged according to the delay margin currently running Whether electric system stablizes, and is as follows：

Step 1: structure time-lag power system mathematical model：It is as follows：

Step 1-1：2, No. 3 generators in 9 node system of WSCC-3 machines are considered as the part containing wide-area control circuit Network equivalence unit is located at the set end voltage measured on No. 2 busbares and there is delay τ during feeding back to field regulator₁, There are delay of feedback τ for the set end voltage measured on No. 3 busbares₂, build the power systems with nonlinear differential algebraic system side containing Time Delay Journey group：

In formula (2)：s∈Rⁿ, it is the reset condition variable of system；y∈R^r, it is the original algebraic variable of system；s_i=s (t- τ_i), i=1,2, it is the original hangover state variable of system；y_i=y (t- τ_i), i=1,2, it is that the original time lag algebraically of system becomes Amount；τ_i∈ R, i=1,2, it is the time lag constant of system；

Step 1-2：By formula (the 2) (x at equalization point_e,y_e) linearisation, it obtains：

In formula (3)：i =1,2；

Step 1-3：Under the premise of not considering unusual, the G in formula (2)_y,G_yiReversible, above-mentioned formula (2) is expressed as：

In formula (4)：

Step 1-4：Indicate that the increment of state variable, formula (3) are rewritten into using x (t)=△ z：

Step 1-5：Time-lag power system mathematical model indicates as follows：

In formula (5)：T indicates time variable；X (t) is state variable；It is state variable to the derivative of time；x(t- τ_i), i=1,2, it is hangover state variable；H (t, ξ) is the historical track of state variable x (t)；ξ∈[-max(τ_i), 0) it indicates Variable ξ is in τ_iChange between the opposite number of maximum value and 0；Above-mentioned algebraic variable belongs to real number field R, and above-mentioned vector variable belongs to In 10 dimension real vector R¹⁰, non-time delay matrix A₀, time delay matrix A₁、A₂Concrete numerical value it is as follows：

Step 2: utilizing time delay matrix A₁、A₂Sparsity, to step 1 structure time-lag power system mathematical modulo Time delay matrix A in type_i, i=1, the sum of 2 carry out Jordan canonical transformations, and are converted according to sparsity procession；Tool Steps are as follows for body：

Step 2-1：To time lag coefficient matrices A_i, i=1,2 sum, and obtain

Step 2-2：To step 2-1'sSummed result carries out Jordan canonical transformations, i.e., to time lag coefficient matrices A_i, i =1, the sum of 2 processions convert：

In formula (9)：T is row-column transform matrix；

Step 2-3：To first formula of the mathematical models of power system containing Time Delay that step 1 is established：It converts to obtain into every trade：

Step 2-4：Variable replacement is carried out to state variable x (t), enables Tx (t)=z (t), then x (t)=T^-1z(t)、x(t- τ_i)=T^-1z(t-τ_i), it substitutes into formula (10) and obtains：

Step 2-5：According to the time delay matrix A in formula (11)_iJSparsity to the time delay matrix A_iJInto every trade Rank transformation, transformation principle are：For some variable z_k, 1≤k≤10, if A_iJIn elements A_iJ(i, j) or A_iJ(j, i), 1≤ I≤2,1≤j≤10, A_iJ(i, j) and A_iJThe value of (j, i) is zero, then by variable z_kMove to state variable sequence afterwards successively most End obtains sequence and rearranges the state variable after arrangement：Wherein, It is obtained according to above-mentioned transformation principle：

In formula (12)：

Time-lag power system mathematical model after Jordan canonical transformations and sparsity row-column transform indicates as follows：

Step 3: utilizing Taylor series expansion hangover state variablesSeparation by Jordan canonical transformations and The non-time lag item in time-lag power system mathematical model after sparsity row-column transformAnd time lag itemBetween it is mutual Association；It is as follows：

Step 3-1：Utilize the hangover state variable in Taylor series expansions (13)

Step 3-2：Formula (14) is substituted into first formula in formula (13)It obtains：

Step 3-3：Similar terms merging is carried out to formula (15)：

It is reversible, the both sides of formula (16) are multiplied by Inverse matrix obtain by Taylor series detach after time-lag power system mathematical model：

Step 4: the time-lag power system number using Schur model reductions to step 3 after the separation of Taylor series It learns model to be simplified, the number of state variable is reduced to 4 by 10, finally obtains the time-lag power system mathematics after simplifying Model；It is as follows：

Step 4-1：It is whole that input-outputization is carried out to the time-lag power system mathematical model after the separation of Taylor series Reason obtains：

In formula (18)：

Step 4-2：10 state variables in formula (18) are reduced to 4 using Schur model order reducing methods：

In formula (19)：z_red(t)∈R⁴, y (t) ∈ R⁴, A_red∈R^4×4, B_red,i∈R^4×4, C_red∈R^4×4,

B_red=[B_red,1 B_red,2]；

Step 4-3：It, will according to formula (18)Substitution formula (19)：

Step 4-4：By the second formula in formula (20)Substitute into the first formula In obtain：

Step 4-5：According to Taylor series, each of input u (t) inputs component u in expansion (18)_i(t)：

Step 4-6：All input components are substituted into formula (21), finally obtain the time-lag power system mathematical modulo after simplifying Type：

In formula (23)：

Step 5: under Power System Delay stability criteria, the time lag power train after the simplification obtained using step 4 System mathematical model finds out the delay margin of the time-lag power system containing 2 Time Delays, to the power train that is guaranteed The permitted maximum time lag of stable operation of uniting.Wherein, the Power System Delay stability criteria include two following situations it One：

(1) Layapunov stability criterias；

(2) Eigenvalues analysis stability criteria.

The effect of this method is illustrated by taking three kinds of specific Layapunov stability criterias as an example below：

Time-lag power system mathematical model after the simplification obtained using step 4 finds out 9 node electricity of double time lag WSCC-3 machines The delay margin of Force system compares by JTS fast solution methods of the present invention and directly with the result of criterion method solution Compared with three typical criterion particular contents of Layapunov stability criterias are as follows：

Criterion 1 provides a kind of electric system multiple time delay stability criteria containing free-form curve and surface.

Criterion 1 is for the system containing m Time Delay, if there is positive definite matrix P ∈ R^n×n, Q_i∈R^n×n, i=1, 2 ..., m, positive semidefinite matrix W_i,j∈R^n×n, symmetrical matrix X_i,j∈R^{(m+1)n×(m+1)n}, 0≤i<j<M and matrix So that following MATRIX INEQUALITIES is set up：

M_i,j>=O, 0≤i<j<m (29)

In formula：

Criterion 2 provides a kind of electric system multiple time delay stability criteria without free-form curve and surface.

Criterion 2 is for the system containing m Time Delay, if there is positive definite matrix P ∈ R^n×n, Q_i∈R^n×n, i=1, 2 ..., m and positive semidefinite matrix W_i,j∈R^n×n, i=1,2 ..., m so that following MATRIX INEQUALITIES is set up：

In formula：

Criterion 3 introduces integral quadratic form, provides a kind of electric system multiple time delay stability criteria of the quadratic form containing integral.

Criterion 3 is if there is positive definite matrix P ∈ R^{(2m+1)n×(2m+1)n}, U_i∈R^n×n, Y_i∈R^2n×2nWith positive semidefinite matrix Q_i∈ R^2n×2n, Z_i∈R^2n×2n, i=1,2 ..., m make lower column matrix negative definite：

System then containing m Time Delay is asymptotically stability, wherein：

A_c=[A₀ A₁ A₂ … A_m-1 A_mO ... O],

C₁=[e₁ e₂ e₃ … e_m e_m+1 e_m+2 e_2m+2 e_2m+3 … e_3m e_3m+1],

Fig. 1 be time lag inhibited stably, Fig. 2 that direct solution obtains be using obtained after JTS fast solution methods when Stagnant inhibited stably.

Comparison diagram 1 and Fig. 2 are, it is apparent that the 9 node power system of WSCC-3 machines obtained by JTS fast solution methods Time lag inhibited stably unite in addition to individual points, the curve almost obtained with criterion direct solution is completely the same.

To compare the time-consuming situation of criterion, defined parameters θ and calculating efficiency increase rate (Calculation Efficiency Increasing Rate, CEIR) it is as follows：

90 ° are increased to from 0 ° according to θ, table 1 gives corresponding calculating time and CEIR values：

The 9 node power system-computed time of 1 WSCC-3 machines of table compares

As it can be seen from table 1 first with 9 node multiple time delay electric power of JTS fast solution methods abbreviation WSCC-3 machines of the present invention System, then delay margin is calculated by stability criteria, it can be greatly decreased and solve the time.When θ becomes between 0 ° to 90 ° When change, the CEIR values of criterion 1 can reach 40131.13%, and the CEIR values of criterion 2 can reach 3266.67%, criterion 3 CEIR values can reach 4613.11%, the results showed that JTS methods have apparent quick effect.

It is former to the rapid solving of JTS methods by taking criterion 1 calculates 9 node power system time lags stability margin of WSCC-3 machines as an example Reason illustrates, after using JTS methods, unknown variable the matrix P, Q of criterion 1₁, Q₂, W₁, W₂, W₃, X₁₁, X₂₂, X₃₃, Y₁₁, Y₂₂, Y₃₃And Z₁₁, Z₂₂, Z₃₃It is 4 × 4 square formations, N₁, N₂, N₃, S₁, S₂, S₃, T₁, T₂, T₃, X₁₂, X₁₃, X₂₃, Y₁₂, Y₁₃, Y₂₃With Z₁₂, Z₁₃, Z₂₃All it is 4 × 4 matrix appropriate, unknown variable sum N₁For：

And for directly utilize criterion solve for, have in unknown variable matrix 15 10 × 10 symmetrical matrix and 18 10 × 10 appropriate matrix, unknown variable sum N₂For：

The use of the unknown variable sum after JTS methods is about relatively directly to calculate and be greatly decreased without using the 1/6 of situation, because This is with higher computational efficiency.The analysis situation of other two kinds of criterions is similar therewith, and the unknown variable of criterion 1 to criterion 3 subtracts Few rate is respectively 83.31%, 81.82% and 83.16%, it can be clearly seen that can be greatly by JTS fast solution methods Unknown variable number is reduced, solution efficiency is improved.

Although above in conjunction with attached drawing, invention has been described, and the invention is not limited in above-mentioned specific implementations Mode, the above mentioned embodiment is only schematical, rather than restrictive, and those skilled in the art are at this Under the enlightenment of invention, without deviating from the spirit of the invention, many variations can also be made, these belong to the present invention's Within protection.

Claims

1. a kind of Power System Delay stability margin fast solution method, which is characterized in that for containing m Time Delay when Stagnant electric system builds time-lag power system mathematical model, using Jordan canonical transformations, the separation of Taylor series, Schur moulds Type depression of order simplifies the time-lag power system mathematical model, the time-lag power system mathematical model after being simplified, into And the delay margin of time-lag power system is quickly found out, to be guaranteed, power system stability runs permitted maximum Time lag is as follows：

Step 1: structure time-lag power system mathematical model：

In formula：T indicates time variable；X (t) is state variable；It is state variable to the derivative of time；A₀For non-time lag system Matrix number；A_i, i=1,2 ..., m are time delay matrix, and m indicates Time Delay number；τ_i, i=1,2 ..., m are system Time lag constant；τ_i＞ 0 indicates that time lag is all higher than 0；x(t-τ_i), i=1,2 ..., m are hangover state variable；H (t, ξ) is The historical track of state variable x (t)；ξ∈[-max(τ_i), 0) indicate variable ξ in τ_iChange between the opposite number of maximum value and 0； Above-mentioned algebraic variable belongs to real number field R, and above-mentioned vector variable belongs to n dimension real vectors Rⁿ；

Step 2: utilizing time delay matrix A_iSparsity, to step 1 structure time-lag power system mathematical model in when Stagnant coefficient matrices A_i, i=1, the progress Jordan canonical transformations of the sum of 2 ..., m, and according to sparsity to time-lag power system number Learn the transformation of model procession；

Step 3: utilizing the hangover state variable after row-column transform in Taylor series expansion step 2Separation is passed through The non-time lag item in time-lag power system mathematical model after Jordan canonical transformations and sparsity row-column transformAnd time lag itemBetween it is interrelated；

Step 4: using Schur models to step 3 by Taylor series separation after time-lag power system mathematical model into Row simplifies, and the number of state variable is reduced to r by n, finally obtains the time-lag power system mathematical model after simplifying；

Step 5: under Power System Delay stability criteria, the time-lag power system number after the simplification obtained using step 4 The delay margin that model finds out the time-lag power system containing m Time Delay is learned, completion Power System Delay is stablized abundant The rapid solving of degree, to the permitted maximum time lag of the power system stability operation that is guaranteed.

2. Power System Delay stability margin fast solution method according to claim 1, it is characterised in that：Step 1 includes Following steps：

Step 1-1：Build the power systems with nonlinear differential algebraic system equation group containing Time Delay：

In formula (2)：s∈Rⁿ, it is the reset condition variable of system；y∈R^r, it is the original algebraic variable of system；s_i=s (t- τ_i), i =1,2 ..., m are the original hangover state variable of system；y_i=y (t- τ_i), i=1,2 ..., m are the original time lag generation of system Number variable；τ_i∈ R, i=1,2 ..., m is the time lag constant of system；

Step 1-2：By formula (the 2) (s at equalization point_e,y_e) linearisation, it obtains：

In formula (3)：i =1,2 ..., m；Δ s, Δ y, Δ s_i, Δ y_iFor the increment near equalization point；

Step 1-3：Under the premise of not considering unusual, the G in formula (3)_y,G_yiReversible, above-mentioned formula (3) is expressed as：

In formula (4)：

Step 1-4：Indicate that the increment of state variable, formula (4) are rewritten into using x (t)=Δ s (t)：

Step 1-5：Time-lag power system mathematical model indicates as follows：

In formula (6)：T indicates time variable；X (t) is state variable；It is state variable to the derivative of time；A₀For non-time lag Coefficient matrix；A_i, i=1,2 ..., m are time delay matrix, and m indicates Time Delay number；τ_i, i=1,2 ..., m are system Time lag constant；τ_i＞ 0 indicates that time lag is all higher than 0；x(t-τ_i), i=1,2 ..., m are hangover state variable；H (t, ξ) is shape The historical track of state variable x (t)；ξ∈[-max(τ_i), 0) indicate variable ξ in τ_iChange between the opposite number of maximum value and 0；On It states algebraic variable and belongs to real number field R, above-mentioned vector variable belongs to n dimension real vectors Rⁿ。

3. Power System Delay stability margin fast solution method according to claim 1, it is characterised in that：Step 2 includes Following steps：

Step 2-1：To time lag coefficient matrices A_i, i=1,2 ..., m sum, obtain

Step 2-2：To step 2-1'sSummed result carries out Jordan canonical transformations, i.e., to time lag coefficient matrices A_i, i=1, The sum of 2 ..., m procession convert：

In formula (7)：T is row-column transform matrix；

Step 2-4：Variable replacement is carried out to state variable x (t), enables Tx (t)=z (t), then x (t)=T^-1z(t)、x(t-τ_i)= T^-1z(t-τ_i), it substitutes into formula (8) and obtains：

In formula (9)：A_0J=TA₀T^-1；

Step 2-5：According to the time delay matrix A in formula (9)_iJSparsity to the time delay matrix A_iJProcession becomes It changes, transformation principle is：For some variable z_k, 1≤k≤n, if A_iJIn elements A_iJ(i, j) or A_iJ(j, i), 1≤i≤m, 1≤j≤n, A_iJ(i, j) and A_iJThe value of (j, i) is zero, then by variable z_kIt moves to state variable sequence most end afterwards successively, obtains The state variable after arrangement is rearranged to sequence：Wherein, It is obtained according to above-mentioned transformation principle：

In formula (10)：

According to above-mentioned steps 2-1 to step 2-5, obtain and time-lag power system mathematical model of equal value in step 1.

4. Power System Delay stability margin fast solution method according to claim 1, it is characterised in that：Step 3 includes Following steps：

Step 3-1：Utilize the hangover state variable in Taylor series expansions (11)

Step 3-2：Formula (12) is substituted into first formula in formula (11)It obtains：

Step 3-3：Similar terms merging is carried out to formula (13)：

It is reversible, the both sides of formula (14) is multiplied by simultaneouslyInverse matrix, obtain by Taylor series detach after time lag power train System mathematical model：

5. Power System Delay stability margin fast solution method according to claim 1, it is characterised in that：Step 4 includes Following steps：

Step 4-1：Input-output is carried out to the time-lag power system mathematical model after the separation of Taylor series to arrange It arrives：

In formula (16)：

Step 4-2：N state variable in formula (16) is reduced to r using Schur model order reducing methods：

In formula (17)：z_red(t)∈R^r, y (t) ∈ R^r, A_red∈R^r×r, B_red,i∈R^r×r, C_red∈R^r×r,

B_red=[B_red,1 B_red,2 … B_red,i … B_red,m-1 B_red,m]；

Step 4-3：It, will according to formula (16)Substitution formula (17)：

Step 4-4：By the second formula in formula (18)Substitute into the first formula In obtain：

Step 4-5：According to Taylor series, each of input u (t) inputs component u in expansion (16)_i(t)：

Step 4-6：All input components are substituted into formula (18), finally obtain the time-lag power system mathematical model after simplifying：

In formula (21)：

6. Power System Delay stability margin fast solution method according to claim 1, it is characterised in that：In step 5, The Power System Delay stability criteria includes one of two following situations：

(1) Layapunov stability criterias；

(2) Eigenvalues analysis stability criteria.